- Find the length of the helix f(t)= [(cost)^2), (sint)^2, 2t^2] for t E [0,10]. Reparametize this curve by arc length.
To find the length you need to integrate
$\displaystyle s=\int_{a}^{b}\sqrt{\left( \frac{dx}{dt}\right)^2+\left( \frac{dy}{dt}\right)^2+\left( \frac{dz}{dt}\right)^2}}}, \quad t \in [a,b] $
This gives
$\displaystyle s=\int_{0}^{10}\sqrt{4\sin^2(t)\cos^2(t)+4\sin^2(t )\cos^2(t)+16t^2}dt}$
This function does not have a "nice" anti derivative.
Did you mean the helix $\displaystyle \mathbf{r}(t)=<\cos(t),\sin(t),2t>$?